INFLUENCE OF FREQUENCY DISTRIBUTION ON INTENSITY FLUCTUATIONS OF NOISE Pierre HANNA SCRIME - LaBRI Université de Bordeaux 1 F-33405 Talence Cedex, France hanna@labriu-bordeauxfr Myriam DESAINTE-CATHERINE SCRIME - LaBRI Université de Bordeaux 1 F-33405 Talence Cedex, France myriam@labriu-bordeauxfr ABSTRACT This article introduces mathematical parameters that generalize the frequency distribution of the statistical and spectral model We show experiments that determine the influence of these parameters on the intensity fluctuations of the synthesized noise, accordingly to the theory developed by psycho-acoustic works The main goal of these experiments is to be able to analyse and synthesize bands of noise with different spectral densities 1 INTRODUCTION Electro-acoustic composers often use real-world or environmental sounds in their works The existing analysis/synthesis models (like for example [1]) can reproduce and transform pseudo-harmonic sounds All these models can be used with non-deterministic sounds with the assumption that everything which can not be represented by sinusoids whose amplitudes and frequencies evolve slowly in time, is represented as filtered (or colored) white noise Limits of these approaches can be clearly heard with applications on natural sounds These sounds are not only very difficult to define mathematically but their perception is not completely understood In this paper we try to give a general form for the frequency distribution of a statistical model and define mathematical parameters The influence of these parameters on the intensity fluctuations are then compared with experiments 21 Spectral density 2 PERCEPTION Intensity fluctuations are stated as a perceptually relevant property of any sound For example, an audible sine tone is perceived as a steady sound whereas it may have many cycles per second This stability is described by the fluctuations of temporal envelope This parameter is very useful to describe bands of noise [2] One main application of the envelope fluctuations is the explanation of the ability for listeners to discriminate sounds with different spectral density [3] Several psycho-acoustic experiments show that this ability can mainly be explained by two cues, one spectral and one temporal The spectral cue corresponds to the pitch associated with the band of noise whereas the temporal one concerns the envelope fluctuations These two parts are the central points of our paper 22 Timbre and roughness Moreover these fluctuations are stated as one of the dimensions of the timbre Indeed they can be related to the complex concept of roughness The roughness is defined by the perceptual effect from the fast beats produced by tones and it is perceived naturally [4] However no theoretical relation exist (to our knowledge) between complex envelope fluctuations and roughness 23 A musical parameter This perceptual parameter also seem to be musically important Schaeffer uses the concept of grain to describe musical differences between sounds [5] It is defined between the rhythm and pitch domains, as many small irregularities at the surface of the sound In this paper, we focus on the dynamic envelope fluctuations of noise and, more generally, on the control of the spectral density In the following, we use the word envelope to describe temporal envelope 3 BACKGROUND A few noise models have already been defined The model we use in this paper is spectral and based on the thermal noise model 31 Thermal noise Thermal noises have been described in terms of a Fourier series [2]: where the pulsations which are equally spaced, (1) is the number of frequencies, is an integer, are are random variable distributed according to a Rayleigh distribution and are random variable uniformly distributed This definition is the starting point of our work 32 SMS model The Spectral Modeling Synthesis (SMS) [1] separates the analyzed sound into deterministic and stochastic parts The main assumption about the stochastic component is that it can be fully described by its spectral envelope It is re-synthesized by inverse-fourier transform This transformation is mathematically described as a sum of a fixed number of sinusoids, whose amplitudes are spectral DAFX-1
C envelope values and whose frequencies are proportional to the ratio The phases are uniformly distributed With this model, intensity fluctuations can only be controlled by modifying the spectral envelope from a temporal window to another: the intensity fluctuations are then related to the spectral envelope (2) N Frequencies L 33 Statistical model A statistical approach have already been presented [6] The sound is described as a sum of sinusoids whose frequencies are random value (with a specific probability density function which is a parameter), phases are uniformly distributed and amplitude are fixed With this model, the intensity fluctuations can t neither be directly controlled 4 STATISTICAL MODEL FOR NOISES In this paper, we consider sounds (sample rate ) in the spectral model as random processes as in [6] Each frequency component is a random variable with fixed amplitude and random phase : (3) where the frequencies are distributed in a band whose width is (Hz) Previous psycho-acoustic works [2] show that phases contribute to intensity fluctuations More precisely, a noise with minimal power fluctuations (low-noise noise [7]) can be synthesized with some fixed phase values [8] Nevertheless in this paper, we focus on the distribution of the frequencies and their contribution to the intensity fluctuations 41 Parameters We propose to generate bands of noise in a statistical way We consider a band of frequencies, divided in bins No more than one frequency can be drawn in each bin We define these parameters (see figure 1): is the number of frequencies which are randomly chosen in a band is the bandwidth (Hz) is the number of bins ( ) is the width of the uniform distribution ( ), centered on each bin In the case of, a bin is first drawn, then a frequency is determined within this bin By choosing infinite, the distribution is uniform By choosing "!, the signal generated follows the thermal noise model Between these two extreme cases an infinity of distributions is proposed In the scope of this paper, L is the same for all the bins But we could easily extend the model and associate a distribution to each bin The spectral density is defined in this paper as 0 Bin center W M Bins 42 Envelope Definition Figure 1: Parameters of the model From the above equation 3, the envelope as or using complex numbers :,+ %$'& '( *) - /103254762%8 + frequency (Hz) can be defined [2] Practically we can easily extract the envelope of any signal from an inverse-fourier transform by removing the negative frequency components from the spectrum This method was used during our experiments The auditory system cannot detect too fast changes of envelope We use a model which take account into this property : the fluctuations are attenuated by a low-pass filter (equation 6) (4) (5) 9 ;: < ->=@? A (6) where < is a time measure in seconds Psycho-acoustic experiments show that < takes value between and B milliseconds [2] We have used B ms during our experiments The envelope is precisely defined in [2] The envelope power is defined by: DC E = GF 43 Envelope fluctuations GF $'& H I JF K) LI ) JFNM Hartmann [3] made psycho-acoustic experiments to understand that the ability to discriminate spectral density is related primarily to the envelope fluctuations These envelope fluctuations are modeled in [2] so that the variance of the envelope power (normalized by the square of the mean) is related to,, and < (equation 8) In his studies, our parameter is defined as and is fixed In the future we could extend this model to take into account other values of the parameter, so that another more complex equation could be defined (7) DAFX-2
:! C : J : I < : I < : L= < < I C I LH : L= < < C M Autocorrelation ratio 10 09 08 07 Autocorrelation Ratio 06 5 EXPERIMENTS (8) 05 04 03 From the model developed by Hartmann, we have decided to measure the influence of the different parameters on the envelope fluctuations by using the variance Several experiments have been performed Two different aspects are tested : the periodicity and the amplitude of the envelope fluctuations 51 Periodicity of envelope fluctuations From the definition of the envelope power (equation 7), one can see that it is periodic This envelope periodicity is related to the frequencies differences These differences can be estimated from the parameters (,,, ) Experiments show that we can hear either a pitch or beats This pitch or beat is independent!g!j! of the spectral!g!j! envelope: a sound generated with : Hz, gives one pitch J!J!G! Hz, centered at :G: at Hz but also a low-frequency pitch or beats corresponding to the envelope fluctuations, in the case of particular spectral density This implies that by modifying our parameters, we can synthesize bands of noise with beats or with pitch Our experiments show that if the number of components increases, the periodicity decreases until low frequency beats appear Moreover if the number of bins or/and the width of the probability density function increase whereas is constant, the periodicity is no more fixed and its standard deviation increases The figure 2 shows the influence of the parameter We have used the autocorrelation function It is known that this function is related to the periodicity perception of any sound For example, in the case of the fixed frequencies (! and ), which corresponds to the thermal noise model, the autocorrelation perfectly shows the periodicity of the synthesized noise We have measured the ratio between the second maximum and the first point of the autocorrelation function (which is the total energy of the signal and the maximum), as a function of the width of the probability density function of the bin This ratio is supposed to be related to the pitch perception The figure 2 shows that increasing decreases the ratio That s why the periodicity (beats) is not as clearly heard and! as in the case of, which corresponds to the thermal noise model and the SMS stochastic model 52 Amplitude of envelope fluctuations We have done experiments to verify the influence of the number of components, the number of bins, and the width of the probability density function In particular, it appears that : Increasing the number of components increases the envelope fluctuations (this can be shown from equation 8 and can be seen figure 3) 02 01 00 00 01 02 03 04 05 06 07 08 09 10 L/(bin width) Figure 2: Influence of the width : autocorrelation ratio as a function of J!G! J!J!G! ( realizations of B samples of a signal with Hz and ) Increasing the width of the probability density function increases the envelope fluctuations (this can be seen figure 3 Increasing the number of bins increases the envelope fluctuations (this can be seen figure 3 and shown from equation 8) The effect of the bandwidth of the noise is related to equation 8 It is obvious that increasing the bandwidth (the number of components being unchanged) results in decreasing spectral density so that envelope fluctuations may decrease or pitch may be heard due to the spectral differences (see figures 6, 4 and 5) 060 054 048 042 036 030 024 018 012 006 000 0 6 12 18 24 30 36 42 48 54 60 sinusoïd number Figure 3: Mean of the of the power envelope as a function of the number of components (B!G! realizations of B >!G! samples of a signal and Hz) when is infinite and and! DAFX-3
B 030 025 020 61 Toward a statistical model These studies may help to define a statistical model which could permit to control independently the spectral and the temporal envelope Moreover the variance of the envelope fluctuations can be used as a measure of the spectral density in order to decide the number of sinusoids and the statistical parameters to draw them, and to synthesize a band of noise 015 62 Windowing 010 005 000 0 35 70 105 140 175 210 sinusoid number Figure 4: Mean of the of the power envelope as a function of the number of components (: realizations of B,!G!G! samples of a signal and Hz) when is and! infinite 0040 and!j!g! This statistical approach may lead to a synthesis model That s why many signals can be synthesized in successive short-time windows The length of the windows are an interesting parameter in the case of periodic audible envelope fluctuations A short window (less than twice the period of the envelope) breaks the periodicity and prevents the user from hearing beats or even pitch This technique is implicitly used in the inverse-fourier transform (SMS model) For a samples long window, this can be represented as the sum of fixed sinusoids (at the center of the bins)this and! One can corresponds to the parameters demonstrate that the envelope fluctuations corresponding to the minimal frequency difference ( ) can not be perceived because of the length of the window Furthermore the statistical approach seems interesting concerning the efficiency To produce a noise with the same envelope fluctuations, or with no audible beat, one should choose and reduce the number of components to 0036 0032 0028 0024 0020 0016 0012 0008 0004 7 FUTURE WORK 71 Noisy sounds analysis and synthesis Applications of the statistical parameters on the intensity fluctuations seem interesting concerning the synthesis of sounds like for example engine-, car-, traffic- or waterfall noises Moreover many experiments may be performed on the analysis and the synthesis of consonants and whispered voices 0000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 sinusoïd number Figure 5: Mean of the of the power envelope as a function of the number of components J!J! ( realizations of J!G!J!G! samples of a signal and Hz) when is infinite and and! 6 APPLICATIONS Many applications may be based on these works about the envelope fluctuations and the influence of the statistical parameters we have defined The first one is an analysis/synthesis model for noisy sounds 72 Phase influence In our works, we have only considered the frequency distribution and we use an uniform distribution to draw the phases of the spectral components (as in the SMS model) Nevertheless we have begun to study the influence of the width of the probability density function of the phase We can see on the figure 6 that increasing this width decreases the envelope fluctuations Noises synthesized with narrow probability density function for the phase are described as impulsive noises Moreover some envelope fluctuations can not be synthesized with a uniform distribution of the phases We will study the limit between a noise with large envelope fluctuations and transients and try to give a precise definition related to our future model As seen previously, very small values of envelope fluctuations are very useful in masking sine signals [9] These small values can only be obtained by choosing phases That s also why this parameter is one of the main point of our future work DAFX-4
B 1000 0833 0667 0500 0333 [6] Desainte-Catherine M and P Hanna, Statistical approach for sound modeling, Proceedings of the Digital Audio Effects Workshop (DAFX 00, Verona), pp 91 96, 2000 [7] J Pumplin, Low-noise noise, Journal of Acoustical Society of America, vol 78, no 1, pp 100 104, 1985 [8] WM Hartmann and J Pumplin, Periodic signals with minimal power fluctuations, Journal of Acoustical Society of America, vol 90, no 4, pp 1986 1999, 1991 [9] WM Hartmann and J Pumplin, Noise power fluctuations and the masking of sine signals, Journal of Acoustical Society of America, vol 83, no 6, pp 2277 2289, 1988 0167 0000 0000 0167 0333 0500 0667 0833 1000 width of phase pdf Figure 6: Mean of the of the power envelope as a function of the width of phase PDF (: realizations of is infinite) samples of a signal and!j!g! 8 ACKNOWLEDGMENTS This research was carried out in the context of the SCRIME 1 project which is funded by the DMDTS of the French Culture Ministry, the Aquitaine Regional Council, the General Council of the Gironde Department and IDDAC of the Gironde Department SCRIME project is the result of a cooperation convention between the Conservatoire National de Rgion of Bordeaux, ENSEIRB (school of electronic and computer scientist engineers) and the University of Sciences of Bordeaux It is composed of electroacoustic music composers and scientific researchers It is managed by the LaBRI (laboratory of computer science of Bordeaux) Its main missions are research and creation, diffusion and pedagogy thus extending its influence 9 REFERENCES [1] X Serra and J Smith, Spectral modeling synthesis: a sound analysis/synthesis system based on a deterministic plus stochastic decomposition, Computer Music Journal, vol 14, no 4, pp 12 24, 1990 [2] WM Hartmann, Signals, Sound, and Sensation, Modern Acoustics and Signal Processing AIP Press, 1997 [3] WM Hartmann, S McAdams, A Gerzso, and P Boulez, Discrimination of spectral density, Journal of Acoustical Society of America, vol 79, no 6, pp 1915 1925, 1986 [4] D Pressnitzer, Perception de rugosité psychoacoustique: d un attribut élémentaire de l audition l écoute musicale, PhD Thesis, Université Paris VI, 1998 [5] P Schaeffer, Traité des objets musicaux, Seuil, 1966 1 Studio de Cration et de Recherche en Informatique et Musique lectroacoustique, wwwscrimeu-bordeauxfr DAFX-5